# Besov-weak-Herz spaces and global solutions for Navier-Stokes equations

## TL;DR

This paper establishes global well-posedness for the Navier-Stokes equations in new, larger critical Besov-weak-Herz spaces, developing foundational properties and exploring solution behavior.

## Contribution

It introduces and analyzes Besov-weak-Herz spaces, expanding the functional framework for Navier-Stokes and related PDEs, with new embedding and interpolation results.

## Key findings

- Proved global well-posedness in BWH-spaces for Navier-Stokes.
- Developed heat semigroup estimates and embedding theorems for BWH-spaces.
- Characterized BWH-spaces as interpolation spaces between Sobolev-weak-Herz spaces.

## Abstract

We consider the incompressible Navier-stokes equations (NS) in $\mathbb{R}^{n}$ for $n\geq2$. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These spaces are larger than some critical spaces considered in previous works for (NS). For our purposes, we need to develop a basic theory for BWH-spaces containing properties and estimates such as heat semigroup estimates, embedding theorems, interpolation properties, among others. In particular, it is proved a characterization of Besov-weak-Herz spaces as interpolation of Sobolev-weak-Herz ones, which is key in our arguments. Self-similarity and asymptotic behavior of solutions are also discussed. Our class of spaces and its properties developed here could also be employed to study other PDEs of elliptic, parabolic and conservation-law type.